6.5.4. Friedman TwoWay ANOVA
Data entry is in matrix format (see 6.0.5. Tests with Matrix Data). Columns selected for this test must have equal number of rows and rows containing at least one missing value are omitted.
6.5.4.1. Friedman ANOVA Test Results
This test is used to determine whether the M samples have been drawn from the same population. Cases are ranked and the mean rank is calculated for each sample. The test statistic is calculated as follows:
where N is the number of rows, M is the number of columns and R is the sum of squares of column rank totals.
The test statistic corrected for ties is:
where K is sum of k3k and k is the number of tied cases for a particular rank. H’ has a chisquare distribution with M – 1 degrees of freedom.
Although the chisquare statistic is commonly used, here we also report an alternative definition of the Friedman statistic based on the F distribution, with M – 1 and (N – 1)(M – 1) degrees of freedom, which is said to produce more accurate results (see Conover, W. J. 1999, p. 301). This version of the Friedman statistic is computed as follows: First the ranks (R_{ij}, i = 1, …, N, j = 1, …, M) in each row and then their column totals are found (R_{j}, j = 1, …, M). The test statistic is defined as:
where:
The test statistic displayed is corrected for ties. The output includes rank sum and mean rank for each variable and correction for ties. The onetail probability is reported using both chisquare distribution with M – 1 degrees of freedom and Fdistribution with N – 1 and (N – 1)(M – 1) degrees of freedom.
6.5.4.2. Friedman ANOVA Multiple Comparisons
If the null hypothesis is rejected as result of the Friedman’s test, then a multiple comparison can be run to find out which column effects are different.
Multiple comparisons with rank sums (TukeyHSD)
Nonparametric Multiple Comparisons are performed in a way similar to the TukeyHSD test using rank sums. The standard error is computed as:
Multiple comparisons with tdistribution
Comparisons are made using rank sums and the tdistribution. The standard error is computed as:
Comparisons against a control group (Dunnett)
If each group of data is to be tested against a control group then select this option. The standard error is computed as follows:
The only difference between the Dunnett test introduced here and the Dunnett test per se is that here the group rank sums are used while the latter uses group mean ranks.
6.5.4.3. Friedman ANOVA Examples
Example 1
Example 10.5 on p. 286 from Armitage & Berry (2002). Clotting times (min) of plasma from eight subjects, treated by four methods are given. The null hypothesis “there is no difference between the four treatments” is tested.
Open NONPARM1, select Statistics 1 → Nonparametric Tests (Multisample) → Friedman TwoWay ANOVA and select Treatment 1 to Treatment 4 (C23 to C26) as [Variable]s. Select Test Results as the only output option to obtain the following results:
Friedman TwoWay ANOVA

Cases 
Rank Sum 
Mean Rank 
Treatment 1 
8 
11.0000 
1.3750 
Treatment 2 
8 
16.0000 
2.0000 
Treatment 3 
8 
23.5000 
2.9375 
Treatment 4 
8 
29.5000 
3.6875 
Total 
32 
80.0000 
2.5000 
Number of Columns = 
4 
Number of Rows = 
8 
Correction for Ties = 
0.0125 
ChiSquare Statistic = 
15.1519 
Degrees of Freedom = 
3 
RightTail Probability = 
0.00169 
F(3,21) = 
11.9871 
RightTail Probability = 
0.0001 
Both tests are significant at the 1% level. Hence reject the null hypothesis.
Example 2
Example 12.5 on p. 278 from Zar, J. H. (2010). A researcher wants to test the null hypothesis “time for effectiveness is the same for all three anesthetics”, or in other words that all means are the same against the alternative hypothesis that they are not all equal.
Open NONPARM1, select Statistics 1 → Nonparametric Tests (Multisample) → Friedman TwoWay ANOVA and include Treatment A to Treatment C (C40 to C42) in the analysis by clicking [Variable]. Check the Test Results output option only to obtain the following results:
Friedman TwoWay ANOVA

Cases 
Rank Sum 
Mean Rank 
Treatment A 
5 
6.0000 
1.2000 
Treatment B 
5 
15.0000 
3.0000 
Treatment C 
5 
9.0000 
1.8000 
Total 
15 
30.0000 
2.0000 
Number of Columns = 
3 
Number of Rows = 
5 
Correction for Ties = 
0.0000 
ChiSquare Statistic = 
8.4000 
Degrees of Freedom = 
2 
RightTail Probability = 
0.0150 
F(2,8) = 
21.0000 
RightTail Probability = 
0.0007 
As the right tail probability is less than 5% reject the null hypothesis.
Example 3
Example 1 on p. 371, Conover, W. J. (1999). A researcher wants to test the null hypothesis “the treatments in blocks (i.e. columns) have identical effects” at a 95% confidence level.
Open NONPARM1, select Statistics 1 → Nonparametric Tests (Multisample) → Friedman TwoWay ANOVA and include Grass 1 to Grass 4 (C31 to C34) in the analysis by clicking [Variable]. Select only the Test Results and Multiple comparisons with tdistribution output options to obtain the following results:
Friedman TwoWay ANOVA

Cases 
Rank Sum 
Mean Rank 
Grass 1 
12 
38.0000 
3.1667 
Grass 2 
12 
23.5000 
1.9583 
Grass 3 
12 
24.5000 
2.0417 
Grass 4 
12 
34.0000 
2.8333 
Total 
48 
120.0000 
2.5000 
Number of Columns = 
4 
Number of Rows = 
12 
Correction for Ties = 
0.0583 
ChiSquare Statistic = 
8.0973 
Degrees of Freedom = 
3 
RightTail Probability = 
0.0440 
F(3,33) = 
3.1922 
RightTail Probability = 
0.0362 
Since the right tail probability is less than 5%, reject the null hypothesis. Therefore, we can proceed with Multiple Comparisons to find out which treatments are different.
Multiple Comparisons with t Distribution
Method: 95% t interval.
** denotes significantly different pairs. Vertical bars show homogeneous subsets.
A pairwise test result is significant if its q stat value is greater than the table q.
Group 
Cases 
Rank Sum 
Grass 2 
Grass 3 
Grass 4 
Grass 1 

Grass 2 
12 
23.5000 



** 
 
Grass 3 
12 
24.5000 



** 
 
Grass 4 
12 
34.0000 




 
Grass 1 
12 
38.0000 
** 
** 


 
Comparison 
Difference 
Standard Error 
q Stat 
Table q 
Probability 
Grass 1 – Grass 2 
14.5000 
5.6434 
2.5694 
2.0345 
0.0149 
Grass 4 – Grass 2 
10.5000 
5.6434 
1.8606 
2.0345 
0.0717 
Grass 3 – Grass 2 
1.0000 
5.6434 
0.1772 
2.0345 
0.8604 
Grass 1 – Grass 3 
13.5000 
5.6434 
2.3922 
2.0345 
0.0226 
Grass 4 – Grass 3 
9.5000 
5.6434 
1.6834 
2.0345 
0.1017 
Grass 1 – Grass 4 
4.0000 
5.6434 
0.7088 
2.0345 
0.4834 
Comparison 
Lower 95% 
Upper 95% 
Result 
Grass 1 – Grass 2 
3.0183 
25.9817 
** 
Grass 4 – Grass 2 
0.9817 
21.9817 

Grass 3 – Grass 2 
10.4817 
12.4817 

Grass 1 – Grass 3 
2.0183 
24.9817 
** 
Grass 4 – Grass 3 
1.9817 
20.9817 

Grass 1 – Grass 4 
7.4817 
15.4817 

Homogeneous Subsets: 

Group 1: 
Grass 2 Grass 3 Grass 4 
Group 2: 
Grass 4 Grass 1 
The overall conclusion is that Grass 1/2 and Grass 1/3 are different.